### Abstract

We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R^{2} be a compact, simply connected set with smooth boundary. We define d_{K} (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d_{K} attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

Original language | English (US) |
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Pages (from-to) | 133-144 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 356 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 2009 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

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## Cite this

Bressan, A., & Wang, T. (2009). The minimum speed for a blocking problem on the half plane.

*Journal of Mathematical Analysis and Applications*,*356*(1), 133-144. https://doi.org/10.1016/j.jmaa.2009.02.039